I'm happy to be corrected, but based on the various contexts in which "pakken" and "nemen" (en "meenemen") are used, it appears to me that "pakken" floats somewhere between "take" and "grab" and "get." It does not necessarily mean the sort of brusque action we associate with "grab" in English. But it is a somewhat simpler action than "take" (which implies more emphasis on a change of location) and a somewhat more specific action than "get."

It seems to me that "pakken" means something like "to take into hand." I'm not necessarily transporting the thing from one place to another (pakken dan meenemen). I'm picking it up. I'm asserting personal control over an object that otherwise is just sitting there. The implication may be that I "brengen" it elsewhere or "meenemen" it, but that's not necessarily the case. Maybe I'm going to stay where I am and use it for some purpose.